Indian Journal of History of Science, 51.1 (2016) 56-82
DOI: 10.16943/ijhs/2016/v51i1/48378
What is Indian about Indian Mathematics?*
P P Divakaran**
(Received 12 April 2015; revised 12 December 2015)
Abstract
The principle and results of mathematics are universal and immutable, or so we believe: the
principles are those which guide logical thought and the results emerge from their applications to
abstractions encoding the world around us, primarily numbers (arithmetic) and space (geometry). There
are, nevertheless, significant variations in the practice of mathematics in different cultures. Where India
is concerned, several cultural traits, spanning its entire geography and history – northwest India in the
earliest Vedic period to Kerala in the 16th century – can be identified. The present article is a preliminary,
largely non-technical, inquiry into the roots of these unifying traits. The most pervasive influence seems
to have been that of an oral and nominal mode of articulating and transmitting knowledge, in other words
that of spoken language. Examples are given to illustrate how the resulting challenges to the doing of
mathematics, generally considered to be an abstract and symbolic science, were accommodated or
overcome, especially interesting being the impact of nominalism on decimal counting and on the notions
of zero and infinity. Other topics discussed include the Indian approach to geometry and its differences
from the Hellenic, the idea of proof in India and its evolution, and the flood of new ideas in 15th and 16th
century Kerala which foreshadowed the modern mathematical mainstream. Some remarks are also offered
on possible mutual influences across different cultures.
Key words and phrases: Cross-cultural transmission; Diagonal theorem; Excluded middle;
Functions and calculus; Orality and nominalism: zero: infinity; Recursion and saskāram; Śulbasūtra
and Euclid; Universality and cultural differentiators; Upapatti and yukti.
1. ONE THEOREM, THREE CULTURES
To begin with, one has to ask what
meaning beyond that of mere geographical
association can be attached to the phrase “Indian
mathematics” in my title. Before attempting to
identify or even speak about whatever it is that is
distinctively Indian in Indian mathematics, we
must first be convinced that such a distinction is a
meaningful one. This is an issue that has not had
a great deal of focussed attention paid to it though
historians of mathematics have often taken
positions, generally subconsciously, on either side
of the divide: ‘mathematics in India’ as opposed
to ‘the mathematics of India’. The former position
accepts that the discipline of mathematics is a
universally identifiable activity of the human
mind, individually and collectively, the finest
attestation of its capacity to reason things through
minutely and precisely. It is the canonical view,
for the very good reason that history largely
vindicates it. Everyone may not be in agreement
on a good characterisation of what this universality
consists in and what its cognitive basis might be,
but we all recognise the beast when we see it. To
take a widely known example (and one which has
special significance for India), the statement that
32 + 42 = 52 is a true mathematical statement,
recognised as such by even those who may not
know why it has a deeper significance than say
something like 3 + 4 = 7. Related to it in no
* Based on a plenary lecture at a Seminar on “Methodological Aspects of Knowledge Production in Pre-Modern India” at the
Centre for Contemporary Studies of the Indian Institute of Science, Bengaluru, 24-26 January 2015.
**Flat No. A11B, Rivera Retreat, The Vara, Ferry Road, Cochin - 683 013; Email: [email protected]
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
immediately obvious way is the equally clearly
mathematical (and true) statement: In a rectangle,
the area of the square whose side is its diagonal is
the same as the sum of the areas of the squares
whose sides are the two sides of the rectangle.
The connection between the two is that, together,
they imply that in a rectangle of sides 3 and 4 units,
the length of the diagonal is a whole number, 5;
or, equivalently, if a triangle has sides of 3, 4 and
5 units, the angle opposite 5 is a right angle. The
geometric statement about areas is the diagonal
theorem (those who are uncomfortable with this
name can replace ‘diagonal’ by ‘Pythagorean’)1
first formulated, in more or less the language I
have employed, in the middle-late Vedic
achitectural/geometric workbooks known as the
Śulbasūtra (ca. 800 BC; Baudhāyana,
Āpastamba). The numbers 3, 4 and 5 are the
smallest examples of triples of whole numbers
which have this geometric property and there are
an infinite number of such triples. Diagonal triples
go back a thousand years earlier and several
thousand kilometers further west, occurring first
on Mesopotamian clay tablets of the Old
Babylonian period, around 1800 BC. The point
for us is that this circle of ideas was seen as true
and good mathematics as much in 19th century
BC Mesopotamia as in 9th century BC India and
6th century BC (or perhaps a little later) Greece.
There are any number of other instances in which
pretty much the same mathematical developments,
ranging from sharply defined individual results
to whole new theoretical structures, have come
up in disparate cultural backgrounds widely
separated in space and time.
It would seem to be obvious then that if
there is sense in invoking cultural qualifiers like
‘Indian’ or ‘Mesopotamian’ in relation to
mathematics, it cannot refer to its substance but
only to such intangibles as the value attached to
1
57
mathematical activity, the manner in which
mathematical discoveries were motivated and
made, thought about, made use of and so on; in
other words, to the cultural matrix in which this
whole body of thought, mathematics, was situated.
The way three different civilisations dealt with the
diagonal theorem and diagonal triples (‘the
Pythagorean paradigm’ in a convenient though
historically misleading short phrase) already
serves as the first sign-posts to what we might look
for. The Babylonians wrote their mathematics
down on durable material from which we learn
that they knew some diagonal triples and also the
geometric theorem for the special case of the
square (which says that the square on the diagonal
of the square has twice the area of the square on
the side): there is a well-known tablet with the
inscribed value of √2, the length of the diagonal,
written in Babylonian sexagesimal figure of a
square with its diagonals and an excellent
approximation of the number notation; there are
even figures of nested squares (on another tablet)
which might have provided a visual demonstration
of its truth. (The choice of the base for counting:
60 in Babylonia, 10 in India and none in Greece,
is itself an interesting cultural differentiator). No
statement of the geometric theorem (even for the
square) has so far been brought to light. General
geometric statements are not a strength of
Babylonian mathematics but there is a remarkable
tablet with a problem requring repeated use of
triples to find a diagonal in three dimensions in
an architectural context.
Pythagoras had a statement of the general
geometric theorem and perhaps a demonstration
of it, according to later, sometimes much later,
Greek sources – their word is all we have for it,
no textual material survived – but he probably did
not have a list of triples. The pre-Euclid history of
the Greek version of the theorem is murky and
No theorem in Indian mathematics is named after an individual, no matter how revered. The diagonal theorem which remained
nameless for a long time eventually came to be called bhujā-koi-kar a-nyāya; kar a is the diagonal of a rectangle or, equivalently, the hypotenuse of a right triangle.
58
INDIAN JOURNAL OF HISTORY OF SCIENCE
what we know about its Greek enunci-ation and
proof is essentially what is in Euclid (4th century
BC). Much had changed in Greek intellectual life
in the preceding two centuries and the Euclidean
transformation of geometry into a rigorously
axiomatic science was the most dazzling of the
triumphs of the resulting deductive approach to
the acquisition of knowledge. This is a very well
known story as is the fact that when Europe
rediscovered its Greek intellectual heritage in the
middle of the second millennium AD, Euclidean
geometry became the embodiment of the ideal to
which all mathematical work aspired. Thus did
the idea of a universe of ‘pure’ mathematics,
sufficient unto itself, come into being and, in
course of time, grow into the vigorous and
apparently limitless enterprise that it is now. The
theorem of Pythagoras remained and still remains
an essential ingredient in many branches of
mathematics – it is the foundation of all notions
of distance in geometry – but for Euclid it was
one among the many results that followed from
his definitions and supposedly universally valid
postulates through the exercise of equally
unquestioned rules of logical reasoning. Euclid
also brought the Pythagorean paradigm to a
satisfactory conclusion: he had a general formula
that produced all triples.
Indian geometry has been from the
beginning less ‘pure’ than its Greek counterpart,
making generous room within it for the
mensurational (the ‘metric’ in geometry) aspects
of the study of space: lengths of lines and,
especially, areas of closed figures. The two facets
of the diagonal paradigm together made for a
perfect setting for the practice of this not-so-pure
geometry or, perhaps, it happened the other way,
the practice came first; in any case the Pythagorean
paradigm set the direction for Indian geometry and
remained its defining theme for as long as it lived.
Already in the first propositions on the subject
(Baudhāyana and Āpastamba), equal weight is
given to the statement of the abstract theorem
(along the lines cited above) and to a collection of
(not too large) triples. In fact one of the later
Śulbasūtras (of Mānava, who probably lived at
about the same time as Pythagoras) has a way of
generating an infinite subclass of triples through
a special case of Euclid’s formula. We also know
that the theorem and the triples were put to a
practical and ritually important use, that of making
right angles in the floor plans of altars (vedi).
The mechanisms of oral transmission of
knowledge in Vedic India were very much more
robust than they were in Pythagorean Greece (and
have remained robust throughout history) and it
is to this fact that we owe the survival of the
Śulbasūtra texts. Their faithfulness to the originals
is not in question, but their highly compressed
sūtra format makes it almost certain that
everything that their authors knew was not set
down, though very likely to have been passed on
down the generations. The old cliché, “absence
of proof is not proof of absence”, is particularly
apt here: there are no proofs in these texts.
Supplying proofs of the diagonal theorem, of
varying degrees of ingenuity and elegance, has
always been a staple of Indian mathematical
history. The simpler ones tend to be
diagrammatically prettier and there is no reason
to doubt the opinion of the pioneering scholars of
these texts that they served as visually convincing
demonstrations of its truth. The compulsions of
the axiomatic method would have made them
unacceptable to Euclid; his own proof is visually
unenlightening, as some of us may remember from
our school days. And that illustrates another of
the cultural differentiators of Indian and Greek
geometry and, more generally, mathematics: what
constitutes a proof, a verifiable demonstration, a
yukti in the sense employed by Nīlakaha, the
great 15th-16th century polymath from Kerala?
The focus in this article will naturally be
on India, with Greek and later European work on
related themes serving as the parallel mathematical
universe for purposes of comparison. It must be
borne firmly in mind that traditional mathematical
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
scholarship in India came effectively to an end
with the arrival of European colonial powers, say
in the second half of the 16th century, at about the
same time as the Greek-inspired mathematical
rejuvenation of Europe took off. Formed as we
are in the extraordinarily fruitful culture that is
today’s mathematical mainstream, it is not easy
for us to keep our modernity aside while trying to
penetrate the minds of the authors of the
Śulbasūtra, or of an Āryabhaa or a Mādhava for
that matter. Moreover, any enquiry into the
unifying ideas that run through the enterprise of
creating, validating and communicating
mathematical knowledge will necessarily intrude
into alien territory: logic and philosophy,
linguistics, cultural history, etc., especially
important in the Indian context being the
primordial role of language and grammar in giving
purpose and form to our mathematics, its Indian
flavour as it were. The notes that follow are to be
thought of as no more than a selective (and largely
non-technical) account, illustrated by the simplest
possible examples, of a very tentative first effort
in that direction.
2. THE
GEOGRAPHY OF
INDIAN
MATHEMATICS
Mathematics in India has a long and
continuous history. Going by textual evidence
alone, the earliest example of mathematical
creativity is the invention of a complete and
systematic method of counting with 10 as the base
(decimal enumeration) and the evidence,
enormous quantities of it, comes from an
unexpected source, the gveda. The gveda has
close to three thousand number names distributed
among all its ten Books, the supposedly early as
well as the later Books I and X (Book VIII, one of
the early ‘Clan Books’, is particularly rich in
them). Postponing a closer look to the next section,
here the author only note two points: i) the almost
perfect regularity of the structure of the number
names can only mean that the decimal system was
well established already by the time the individual
59
poems of the early Books were composed; and ii)
the consensus among Vedic philologists is that the
final redaction of the gveda in the form in which
it is known today more or less – via Śākalya,
Sāyaa, et al. – happened around 1200 - 1100 BC
and that the earliest individual poems are to be
dated a few centuries prior to it. The two points
together mean that the perfecting of the decimal
number system took place in tandem with the final
stages of the evolution of Vedic grammar and the
mathematically more interesting invention of the
rules of metrically ordered verse. As for when and
where: around 1400 BC give or take a century,
wherever in northwest India the Vedic people were
at that time.
The earliest Śulbasūtras were compiled
further east, in the Kuru country around Delhi.
After that there was a long break in significant
mathematical activity until we come to the last
centuries BC which saw a tremendous revival of
interest in numbers and their potential infinitude,
especially (but not only) among Jainas and
Buddhists. To roughly the same period belongs
Pigala, whose combinatorial investigation of
prosody began with Vedic metres and ended up in
the complete theoretical classification of all
possible metres in a syllabary with two durations,
guru and laghu, for each syllable, an achievement
of the highest order. We do not know where he
lived.
Around the 3rd century AD, the nature of
mathematical activity began to change. Contacts
with the Hellenic world, especially Alexandria in
Egypt, brought into India the quantitative study
of astronomical observations and the making of
geometric models of planetary motion. This was
a first; the idea of subjecting natural phenomena
to mathematical analysis – the very foundation of
modern science – whose results in turn could be
used to predict events which had not yet come to
pass was not part of Indian intellectual tradition
and, later, it never was extended to other sciences
than astronomy. In any case, astronomy became
60
INDIAN JOURNAL OF HISTORY OF SCIENCE
the driving force behind mathematical innovation
to a very great extent, a privilged role it retained
until the end. The Greek influence probably
entered through Gandhara along with much else
that was Greek but took firm root in the Malava
region during the 4th and 5th centuries (the
Siddhanta period, on account of several competing
astronomical systems or siddhāntas). The last
Siddhāntist was Varāhamihira (Ujjain, 6th century)
but just before that Āryabhaa had already
announced himself at Kusumapura in Magadha
far to the east, where he composed his eponymous
masterpiece bringing together Alexandrian
astronomy and Indian geometry in a remarkable
synthesis. The Āryabhaīya (499 AD) defined the
direction that mathematical astronomy and
mathematics itself were to take in India, much as
Pāini’s Aādhyāyī did for Sanskrit grammar and
grammatical theory in general.
The Aryabhatan revolution inaugurated a
period of extraordinarily rich and diverse
mathematical achievements and it lasted until
creative mathematics came to an end in a blaze of
glory, in Kerala. In between, outstandingly good
mathematicians popped up in virtually every
region of cultural India. To name only the most
influential, we have Bhāskara I and Brahmagupta
(both 7th century) in western India and Bhāskara
II (Bhāskarācārya) in the northern Deccan (12th
century) sandwiching between them some fine
mathematicians working in various parts of north
India; Mahāvīra (9th century) and Nārāyaa (14th
century) in the southern Deccan; the Chera royal
astronomer Śañkaranārāyaa and his teacher
Govindasvāmi (9th century) from almost the
southern tip of Kerala; and finally the remarkable
school established by Mādhava of which
Nīlakaha and Jyehadeva were prominent
members, also in Kerala but further north in the
Nila river basin (14th - 16th centuries). The
centuries after Āryabhaa also saw the first
documented transmission of mathematics from
India to the Baghdad Caliphate, to China and to
southeast Asia; the first surviving examples of
positionally written decimal numbers, including
a zero symbol in the form of a small circle, are in
Sanskrit inscriptions from Cambodia (7th
century).
One purpose of my running through the
names of the chief protagonists in the story of
Indian mathematics and their theatres of action,
however sketchily, is to provide background for
what follows: the names will help set the scene.
The overriding impression, however, is not of how
they differed one from another but how closely
they were linked, over this vast span of time and
geographical distance, by a sense of continuity in
conceptual framework and technical apparatus.
There is an identifiable DNA and it can be traced
all the way back, in spite of the many mutations
along the way such as those brought about by the
great breakthroughs of an Āryabhaa or a
Brahmagupta or a Mādhava. To cite only two out
of many examples, while Āryabhaa’s
trigonometry and Brahmagupta’s geometry of
cyclic quadrilaterals are quite distinct in their goals
and the means by which they were attained, their
seeds are clearly discernible in the Śulbasūtra. The
second example, even more dramatic, is
Mādhava’s invention of calculus for trigonometric
functions: the trigonometry is Āryabhaa’s but the
crucial infinitsimal input relies on a much earlier
idea, the recognition that numbers have no end.
There are excellent historical reasons for
this uniformity of thought and method.
Throughout histoy Indian mathematicians, like
many other Indians – traders, pilgrims and
proselytisers of new faiths, learned people,
architects and artisans, soldiers and those seeking
to evade them, etc. etc. – took to the road readily,
often over very long distances, and the knowledge
in their minds travelled with them. Plenty has been
written about the gradual movement eastwards of
the Vedic people, occupying most of the Ganga
basin by the time of the Buddha, but Āryabhaa
was not descended from that stock. He was from
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
Aśmaka (“aśmakīya” and “aśmaka-janapadajāta” according to Bhāskara I and Nīlakaha)
some thousands of kilometers to the west; the
plausible scenario is that he or his immediate
ancestors were refugees from the invading Hunas,
part of a great mid-5th century migration that also
carried Mahayana Buddhism from its stronghold
in Gandhara and thereabouts to less endangered
areas of Gupta India, in particular eastern India.
Bhāskara II’s birth was near the modern town of
Chalisgaon in northern Maharashtra but there is
an inscription put up there by his grandson tracing
the family back five generations to a paternal
ancestor in the court of Raja Bhoja of Dhar – who,
we may recall, fought Mahmud of Ghazni – in
Malava. There is in fact a striking absence of
‘published’ mathematics (or astronomy)
attributable to north India after the Afghan
invasions and until the end of the 16th century.
Mathematical learning became confined, by and
large, to southern India.
But it was the same mathematics. Well
before the time of the Afghan raids,
Govindasvāmi, a devoted follower of Bhāskara I,
and through him of Āryabhaa, was in Kerala,
surely as part of the early wave of Brahmin settlers
brought to the coast of Karnataka and Kerala by
local kings and chieftains. The identified sources
of these migrations are Ahicchatra on the Ganga
and Valabhi in Gujarat where Bhāskara I probably
taught. And the fact that Mādhava was a Tulu
Brahmin, a recent arrival in the Nila valley from
coastal Karnataka, itself a place of settlement of
northern Brahmins, only serves to reinforce the
northern connection further.
Textual and other internal evidence, as
much as the mathematical content, fully vindicates
the idea of a community of scholars scattered in
various parts of India at various times but
connected together in a permanent intellectual
2
61
network, holding on to their traditional knowledge
systems and the language, Sanskrit, in which the
knowledge was expressed and preserved. Every
student, no matter where he lived, began his
studies with Sanskrit and its grammar before
moving on to the specialised texts composed by
his great predecessors, the pūrvācārya, no matter
how far back in the past or how re-mote
geographically. Despite its origins in Vedic ritual,
the geometry of the Śulbasūtra was the common
heritage of Hindu, Buddhist and Jaina
mathe-maticians alike. Varāhamihira in Ujjain
became thoroughly conversant with Āryabhaa’s
work in distant Kusumapura within about thirty
years. A thou-sand years later, the hold of
Aryabhatan ideas on the Nila mathematicians was
so strong that some historians refer to them as the
Āryabhaa school. One can cite any number of
other instances: the past was part of the living
present, at all times and everywhere. No Indian
mathematician had to redis-cover the work of
pūrvācāryas as Europe had to rediscover the
achievements of the Hellenic civilisation before
laying claim to its intellectual inheritance.
3. ORALITY
AND ITS LEGACY: THE GRAMMAR
OF NUMBERS
Going through an Indian mathematical text
is a disorienting experience for a modern reader
who has not been exposed to one earlier. It is all
words, with no symbols and no equations; even
the numbers are spelt out in words, as their literal
names or linguistic substitutes for them.2 This is
as true of works composed in cryptic sūtras as in
more expansive verse or even more dilatory prose.
We believe we know how the trend started: one
cannot have a symbolic notation without writing
and the scholarly consensus is that the Vedic
culture was an oral, verbal culture which did not
have writing or at least had no use for it when it
There are also very few diagrams in the manuscripts, generally written down fairly late, but that is probably because of the
difficulty of drawing reasonably precise figures on palm leaf with a metal stylus. There is good evidenc that figures formed part
of doing and teaching geometry.
62
INDIAN JOURNAL OF HISTORY OF SCIENCE
came to the sacred; and the sahitā texts, as much
as the instructions for the making of vedis – and,
hence, geometry itself – were sacred. That the very
early mathematics was expressed nonsymboloically is then not a surprise; the surprise
is that the spurning of the symbolic and the written
survived the arrival of writing – using, it may be
added, a systematically organised, abstract and
very symbolic syllabary. Perhaps the responsibility
lay with the very conservatism that ensured
mathematical continuity. The fact is that the
reliance on the oral had enormous consequences,
not only in the communication of mathematics but
also in its exposition and in the way it was thought
about.3
To begin at the beginning, the greatest
influence of orality, mathematically, was on the
genesis and evolution of a system for counting,
that most fundamental of all mathematical skills.
Numeracy, especially in its symbolic form, is now
such a routine accomplishment that we barely
pause to wonder at the sophistication of the
principles that make the act of counting – and
everything else that depends on it including much
of mathematics – possible at all. At its root lies
the idea of a base, 10 in India, but in principle any
other – the 60 of the Babylonians for example –
will do as well. Staying with the familiar decimal
base, the role of 10 in apprehending precisely a
large number is similar to that of say a unit of
length, the meter, in measuring an otherwise
unknown length. The basic arithmetical operation
that applies to such measurements is division with
remainder. Thus, take a number and divide it by
10, say 1947 = 194 x 10 + 7 and repeat the
operation until the quotient becomes less than 10:
1947 = 194 x 10 + 7 = (19 x 10 + 4) x 10 + 7 and
so on, resulting finally in 1 x 103 + 9 x 102 + 4 x
101 + 7 x 100. The problem of the cognition of any
3
number whatsoever is thereby reduced to that of
numbers less than 10 that are the coefficients of
powers of 10 which I will call, following
Bharthari (see below), atomic numbers. The key
to the written representation of a number is in the
realisation that these coefficients, specified in
order, define the number uniquely; thus the
sequence (1, 9, 4, 7), ordered to the right in
decreasing powers of 10 and abbreviated to 1947,
is shorthand for a polynomial when the variable
is fixed at 10 (the coefficients are necessarily less
than 10). The rules of arithmetical operations are
all reflections of corresponding algebraic rules for
polynomials though historically, of course, it
happened the other way, from numbers and
arithmetic to algebra.
The description above of the mathematics
of enumeration is tailored to a written symbolic
representation of numbers. The only extra
ingredient it needs to be completely unambiguous
is a symbol to mark an empty place, in other words
a zero symbol as in 409 or 490. What can one do
to carry out the same construction of numbers
when one has no writing? The Indian answer is
that one gives them spoken names. Begin with
names for the atomic numbers, arbitrary but
unanimously agreed, eka, dvi, . . ., nava, just like
the arbitrary symbols 1, 2, . . ., 9. Then one assigns
names to the places, the positions corresponding
to the ordered powers of 10, also in principle
arbitrary (although they may have – and some do
– other meanings than numerical): daśa, śata, etc.
Finally one combines these two sets of names to
arrive at an ideal naming system for every number
however large, inventing new names for higher
and higher powers of 10 as the need is felt. But
we are now dealing not with an abstract
convention but a living language, Vedic Sanskrit,
and the language has a grammar, and the grammar
There is a partially preserved text, the Bakhshali Manuscript, that has a fair amount of symbolic notation and it is the only such.
It is a singular document in other ways as well, not easy to fit into the standard chronology (4th or 5th century AD in my view).
At least some mathematicians probably employed symbolic notation ‘behind the scenes’. If they did, they kept it out of their
formal writing, and it is difficult to know how widespread the custom was and whether there was anything like a uniform
notational convention.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
has rules for combining words so as to create other
words with a specific, grammatically determined,
meaning. Forming number names is a small part
of the function of such general rules of nominal
composition but they may not be violated. They
must also have the universality and precision that
mathematics demands: universal meaning that the
rules should be categorical, not depending on the
individual numbers being combined, and precise
in the sense of the result being a uniquely
identifiable number, conditions that are not
indispensable in general linguistic usage.
There is no manual devoted to the science
of numbers in the Vedic literature, perhaps
because, unlike geometry, decimal counting and
its evolution were organically part of the
systematisation of Vedic grammar itself. In fact
even the mathematical texts, over the centuries,
pay little attention to the subject but for the
numerous listings of the names of powers of 10.
(A partial exception is the last great masterpiece
of Indian mathematics (in Malayalam prose),
Yuktibhāā of Jyehadeva (ca. 1525)). The
absence of a theoretical tract is more than made
up for, fortunately, by the abundance of numbers
in several Vedic texts, with the gveda at the head,
of which many are compound numbers – those
which are neither atomic nor powers of 10 –
requiring the application of the rules of nominal
composition in the formation and analysis of their
names. For an appreciation of the the kind of
analysis the decryption of a compound number
name calls for, it helps to keep two facts in mind,
one primarily mathematical and the other
grammatical. Firstly, there are only two operations
involved in making a polynomial, multiplication
and addition (in logical order), which are to be
encoded into the grammar of number names: thus
409 = 4 × 100 + 9. Grammatically, addition is
4
63
conjunction and though there are several samāsas
by which it is implemented, they are easy to spot.
Secondly, short of saying that every other other
kind of composition represents multiplication, an
independent grammatical analysis of
multiplicative composition is more involved. The
key here is that most numbers function as
adjectives, so many cows for example; in ‘four
hundred cows’, ‘four hundred’ is an adjective and
‘four’ is an adverb qualifying the adjective
‘hundred’, signifying the 4-fold repetition of the
action of counting up to 100. The rules for the
formation of such repetetive numerical adverbs
in the gveda can be extracted and they are
basically the same as codified by Pāini much later
– allowance being made for some chandasi
exceptions – as are the rules for their declensions
via various affixes. With two or three irresoluble
exceptions, every number name yields a unique
number when Pāini is supplemented by an
occasional appeal to the older authority of the
word-for-word reading, the padapāha.4
The evidence of the gveda is doubly
important in tracing the antiquity of decimal
enumeration because there has always been a
subconscious tendency, natural in view of the
ubiquity of writing, to confuse the abstract
principle with its symbolic manifestation. As far
as the principle is concerned, the symbolic and
the nominal are just equivalent representations of
an abstract structure.
4. ORALITY
AND ITS LEGACY: BEYOND
COUNTING
The symbiosis between numbers and
grammar touches only one – undeniably the most
important – facet of how orality impacted
mathematics (as it did many other fields of
But the analysis is not always trivial. In the 19th century, before H. Kern’s first published edition of the work came out in 1874,
the Āryabhaīya was known to Western scholars only by name as Āryāaśata (for the 108 verses in its three substantive chapters), which is how Brahmagupta referred to it. H. T. Colebrooke, the first translator of Brahmagupta, and E, Burgess, ditto of the
Sūryasiddhānta, both read aśata as 800, a mistake they would not have made had they paid attention to how repetitive adverbs
(āvtti-vācaka) are to be formed.
64
INDIAN JOURNAL OF HISTORY OF SCIENCE
activity). A receptive ear can hear its echoes
resonating through the centuries, all the way down
to the last significant mathematical writings of
16th century Kerala. Let us turn now to a brief
and highly selective recapitulation of some of
these aspects.
The most striking at first sight (one should
perhaps say at first hearing), is mathematical
language. With a few exceptions, texts were
composed in (Sanskrit) verse, the earlier ones,
including the Āryabhaīya, in an almost
impenetrably dense sūtra format. Orality puts a
premium on memory. In the absence of written
texts, knowledge exists only in the memories of
those who possess it and the resort to metrical
verse, like the extreme concision, was an essential
aid to memorisation. Versification in its turn
brought up interesting and difficult mathematical
questions and it took a long time for them to be
precisely formulated and resolved, by Piñgala in
his Chandasūtra (3rd or 2nd century BC?).
Deprived of writing, language is the articulation
of sounds and phonetic accuracy is the guarantor
of its integrity. The regularity of organisation of
the Sanskrit syllabary, already clearly evident in
the earliest Vedic texts, but given its final form at
about the time of Pāini, is part of the response of
the linguistsages to the challenge of preserving
textual purity. Given that every syllable falls into
one of two subsets depending on the duration of
its articulation, the sequencing of syllables in a
line of verse will determine a quasi-musical pattern
(aided in Vedic recitation by the accents, which
disappeared from later, classical, Sanskrit,
probably with the arrival of writing) which is what
a metre is. Piñgala characterised all such patterns
for any syllabic length, thereby shutting the field
down for good and, in the process, initiating the
discipline of combinatorial mathematics. It is an
astonishing piece of work; nothing like it existed
in any other civilisation and we can see why: it
needs a phonologically structured syllabary such
as that of Sanskrit for a mathematical study of
prosody, the science of chandas, to be at all
feasible. The other remark is that Piñgala’s formal
methodology owes something to Pāini – that is
not surprising; he was likely a near contemporary
of Pata–jali – particularly in the use of
metamathematical/metalinguistic labels to
designate subsets of objects sharing a common
mathematical property or following a common
linguistic rule. The curious fact is that Indian
mathematicians seem to have paid him little
attention until quite late and never to have made
use of the notion of a set.
Piñgala comes into the picture in another
role which has nothing directly to do with
combinatorics but ties up with orality in a different
way. He was the first to refer to the zero, śūnya,
as a mathematically defined object, a number like
any other. The idea of nothing, an emptiness, a
vacancy, has been a subject of endless fascination
to Indian learned men past and present, from
grammarians to philosophers, going back to Pāini
and his much-cited aphorism, adarśanalopa,
“that which is not present is lopa”. The story of
the mathematical śūnya is more mundane. The
gveda does not have it nor does any other text
before the Chandasūtra. And it occurs there in a
context which makes its strictly numerical
connotation absolutely clear: it is paired with dvi
as a metamathematical marker, names of subsets
occurring in a particular metre-classification
problem.
The reason for the late appearance of zero
as a number, a thousand years after the gveda
and its profusion of numbers, is not difficult to
see: an oral, nominal system of enumeration does
not need it, unlike a written, symbolic system
which, as we saw, cannot do without it; there are
natural linguistic alternatives like ‘there is no cow’
to the unnatural ‘there is zero cow’. Nor does
arithmetic require its use. To multiply 409 and 51
for example, it is enough to say: (4 hundreds and
9 ones) times (5 tens and 1 one) is (4 times 5)
thousands and (4 times 1) hundreds and (9 times
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
5) tens and (9 times 1) ones and go on from there,
resulting in the answer (2 ten thousands and 8
hundreds and 5 tens and 9 ones) (= 20859 in
symbols), with not a śūnya in sight.5
More puzzling is the absence of a symbolic
zero after writing became widespread. The earliest
known examples of writing in India, the edicts of
Aśoka, have numbers in them but no zero. It is
puzzling because Aśoka and Piñgala lived within
maybe a century of each other. Even more
mystifyingly, there is no zero in any of the many
inscriptions from the following several centuries,
on stone and coin, generally in Brāhmi characters,
though they have lots of numbers. We have already
referred to the fact that the first known, dated,
symbolic zeros are from 7th century southeast
Asia. Even if the zero in the shape of a dot
(śūnyabindu) in the Bakhshali manuscript is
earlier, perhaps as early as the 4th century, that
still leaves a gap of about seven hundred years
between Piñgala’s mathematical zero (and the
advent of writing) and its first written
manifestation. What can account for this long
delay in the passage from abstract idea to concrete
symbol of a concept which, for many, is the very
essence of the (decimal) place-value system?
Part of the answer is that it is no such thing;
people could manage very well without it, and did,
for a long time. The clue to the mystery of the
missing zero really lies in that fact. The written
Brāhmi number system went through a phase of
experimentation before acquiring some sort of
regularity, say by about the 2nd century AD (in
the inscriptions of Nasik for example). It had
symbols for the atomic numerals as well as
individual single symbols for 10, 100 and 1000
(no zeros there), and these were combined to form
multiples of powers of 10 by attaching atomic
numerals as suffixes to the powers of 10: it is as
though we were to choose to write 300 and 400 as
5
65
1003 and 1004 (remember that 100 itself was a
single symbol). The additive part of the
polynomial algorithm is then implemented by
writing the terms to be added side by side, like
10049 for 409. We can recognise this strange
system, symbolic but not positional, for what it is
right away. It is none else than a faithful rendering
into symbols of the names of atomic and powerof-10 numbers, and also of the grammatical rules
that bind them together, from the pre-existing
verbal nomenclature; suffixing and juxtaposition
are, symbolically, multiplicative and additive
composition respectively. And, naturally, there is
no 0 because it is not needed. Brāhmi numbers
are visual symbols not for the numbers themselves
but for their grammatically determined names;
they really are a testimony to the absolute primacy
of the spoken language.
We have no idea when written numbers
finally freed themselves from the tyranny of orality
and became fully positional. Aside from the
undated Bakhshali manuscript, the first dated
written positional numbers are from the end of
the 6th century. They do not have a zero but that
is probably a matter of chance and, in any case,
less important in the grand scheme of the evolution
of numeration than that they are truly positional.
It is this scheme that then travelled to Persia and
the Abbasid kingdom and eventually to Europe.
The pervasive influence of language and
the resistance to symbolic notation had other
consequences such as the premium put on the
precision of technical terminology. The fact that
Sanskrit is a language that values syntactical rigour
above all meant that technical terms, very long
when necessary, could be manufactured with a
high degree of specificity, mitigating to a limited
extent the disadvantages of a narrative style of
doing mathematics. But, without a matching
degree of linguistic (and mathematical) sensitivity
This is trivial but it had to be said. What it reflects is the fact that no 0 is required in the usual way of writing polynomials nor in
algebraic operations with them; a term with coefficient 0 is just absent, lopa in the Pāinian sense. The 0 symbol is the price to
be paid for abbreviating the full polynomial as it is normally written to the sequence of its coefficients as in a written number.
66
INDIAN JOURNAL OF HISTORY OF SCIENCE
on the part of the auditor or reader, it could also
lead to misunderstanding and error. Examples of
how the original exactitude of nomenclature was
lost in sloppy readings are easy to find but let us
restrict to one, historically significant, which has
only recently been cleared up. Of the major
achievements on which Brahmagupta’s greatness
rests, one is the creation of an elaborate theory of
cyclic quadrilaterals, four-sided figures whose four
corners lie on a circle. It is a remarkable body of
work which took its inspiration from the
Śulbasūtra and the diagonal theorem (the
continuity mentioned earlier) and carried it to a
spectacular conclusion. But the mathematics is not
the issue here. Brahmagupta invented the term
tricaturbhuja for the object of his interest and it
caused much confusion among later scholars
including some famous traditional commentators
(presumably not descended from him through a
line of oral transmission) – what could a 3-sided
4-sided figure be? – who chose to interpret it as a
conjunctive compound: a trilateral and a
quadrilateral, a reading not supported by either
mathematics or grammar. The confusion
disappears once it is realised that Brahmagupta’s
theory of the cyclic quadrilateral is built on an
imaginative and powerful exploitation of one key
property: while it has four sides, only three can
be assigned independent lengths, the fourth then
being fixed; a cyclic quadrilateral really is a
triquadrilateral in this sense. Grammar vindicates
this reading.6
truth, that all of it, even theorems as far removed
from being self-evident as can be – think of the
nine-point circle – follows relentlessly from a
handful of apparently self-evident postulates, with
an equally self-evident set of rules of reasoning
as guiding principles. The Elements was the first,
and for a very long time the only, mathematical
work in which every result emerged out of the
inexorable logic of deductive reasoning, without
room for ambiguity or doubt, the perfect example
of a deductive system of knowledge.
5 GEOMETRY
Indian geometry is much more visual than
that of Euclid, in the sense that diagrams convey
a good idea of the geometric truth behind them
(as some of the figures below illustrate), rather
than just serve as props in the logical
argumentation. The reason probably is that it has
generally stayed in touch with the real world: it
began in the architecture, streetscapes and
decorative plastic arts of the Indus Valley – an
When it comes to geometry, we are all
Euclideans. Euclid defines geometry for us as the
decimal system defines numbers, we know no
other kind. It bears recalling why. First comes
admiration for the magnificent edifice he erected,
proposition built upon proposition, easy or
difficult, and then the realisation of the deeper
6
For anyone brought up in this world of
mathematical certitude (in both senses), the first
encounter with the geometry practised in India,
from the Śulbasūtra onwards, can be
disconcerting. There are no lists of postulates, in
fact there are no postulates. Their role is taken
over by (unspecified) notions on which everyone
(presumably) agreed – a common store of
knowledge – such as: two lines each perpendicular
to a given line will not intersect. Definitions are
implicit in the constructions, not spelled out;
nowhere does one find a circle, drawn with a pair
of cord-compasses, defined as the locus of points
at a constant distance from a fixed point. Educated
common sense takes the place of Euclid’s
“common notions”. Altogether, rather than the
strictly segregated categories of unquestionable
assumptions Euclid needed to get going, we find
a more fluid foundation of intelligent good sense,
unquestioned at a given time but not
unquestionable.
There is a verse in the Āryabhaīya which refers to trilaterals and quadrilaterals together. The term employed is the conjunctively
correct tribhujāccaturbhujā.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
area whose study is still in its infancy – and it
retained its architectural bias in Vedic times before
becoming the mathematical vehicle of astronomy
with Āryabhaa. The basic building block is the
circle, easy to draw with the cord-compass, two
sticks and a length of cord stretched taut between
them. Circles lead naturally to pairs of mutually
perpendicular lines in several different ways. The
one that found favour in India, already in the Indus
artefacts and right through till the end, arises as
follows: when two circles intersect, the line
conneting the two points of intersection, the
common chord, is perpendicular to the line joining
the centres; if the circles are of equal radii, the
two lines are, in addition, divided equally by their
point of intersection. There is a method for
constructing a square with the cord-compass at
the beginning of the earliest Śulbasūtra, that of
Baudhāyana, which uses nothing but this property
(which I will call the Indian orthogonality
property). Rather than reproduce Baudhāyana’s
instructions for drawing it (see Fig. 1), I will let
the reader’s eye wander over the various circles
and the lines joining their intersections, take in its
visual appeal and then let the geometric truth
emerge. (For the unconvinced, it is an instructive
exercise to prove, starting from Euclid’s axioms,
that ABCD is in fact a square).
The construction is (or should be)
something of a pivotal point in any enquiry into
the Indianness of Indian geometry. Historically,
sequences of intersecting circles generating
Fig. 1. Square from circles (no diagonal theorem).
67
pleasing patterns, of which the square in Fig. 1
represents the ‘unit cell’, occur in quite a few Indus
Valley artefacts. The similarity is so uncanny that
it is not possible to doubt that Indian orthogonality
represents a direct line of continuity between Indus
and Vedic cultures, one of the very very few such
tangibles that we have. It is also quite unGreek.
In Euclid’s austerely deductive geometry, the
circle comes late – its first appearance is in Book
III – well after what for him are the elementary
building blocks, lines, and the constructs that
follow immediately, their intersections, angles
between them, figures composed of them,
especially triangles (including the Pythagorean
property of right triangles). The most fundamental
relationship between a pair of lines is
perpendicularity in India, not parallelism as for
Euclid, so much so that the term tribhujā always
meant a right triangle rather than a general one.
One can cite many such oppositions. Taken
together, they are a good reason to be cautious
when questions of mutual influences come up as
they have, repeatedly, over more than a century.
Mathematically, the prime significance of
the ‘square from circles’ construction is that the
diagonal theorem and diagonal triples play no part
in it. Indeed, there is no sign of the Pythagorean
paradigm in any of the (admittedly few) places
we might expect to find it among Harappan
remains, including in the perpendicular street
grids; Indian orthogonality is enough. In the
Śulbasūtra, on the other hand, it is omnipresent.
Where did it come from – indigenously discovered
or through contact with the intervening
Mesopotamian civilisation? We will return to this
question in section 9 below but, for the present,
only note that we have good reasons to believe
that the Vedic sages knew the diagonal theorem
to be a theorem; it was not guesswork, they had
proofs.
Before that, let us turn briefly to another,
simpler, construction from the Śulbasūtra which
also has a link with the Indus civilisation as well
68
INDIAN JOURNAL OF HISTORY OF SCIENCE
as with an equally far off future development,
namely Āryabhaa’s trigonometry. It is a method
(the only one documented in India) for
determining the prācī, the true east independent
of location and season, by means of the shadow
of a vertical rod: fix the rod at the centre of a circle
and mark the two points at which the tip of its
shadow touches the circle (W and E in Fig. 2)
during the course of a day; then WE points to the
east.
Fig. 2. Determination of the prācī.
The plausible continuity of geometric
themes of which we have already seen an example
argues for this simple method as having originated
in the cardinal orientation of streets and buildings
in the Indus Valley cities. The connection with the
trigonometry of the future is equally unmissable:
it is impossible to look at the figure without seeing
at once that the north-south line – also needed by
the ritualists – cuts the chord EW perpendicularly
into two equal halves (Indian orthogonality,
effectively), each of which is the half-chord
(jyārdha) or the sine. All of trigonometry is
encapsulated in this diagram and in the diagonal
theorem applied to the right triangle of which the
half-chord is one side. The half-chord as a concept
does not occur in Greek geometry. We might say
that trigonometry as a discipline has its distant
origin in the needs of ritual architects; it is very
Indian in the way it is rooted in practical needs, is
easily visualisable and brings together two of the
signature elements of Indian geometry, the circle
and the diagonal theorem. The direction east
retained its special role as the axis of reference
even after geometry lost its sacral moorings, in
the choice of celestial coordinates in astronomy
as well as in pure geometry where it became the
equivalent of the modern positive y-axis. More
generally, from a cultural perspective, geometry
seems to have undergone two contextual
transformations: from secular (Indus Valley, as far
as we can tell; so very little is known about its
ritual practices) to sacred (Śulbasūtra; perhaps an
example of the bestowing of magical powers on
newly discovered knowledge) and back to secular.
That each step in the evolution took such a long
time, we might also say, is just another signature
of Indianness.
The Śulbasūtra of course have no proofs.
As far as the diagonal theorem is concerned, a
collection of them have been supplied over the
centuries by various commentators. The simplest
are direct and visual, based on cutting up the
relevant areas appropriately and refitting the pieces
differently; in Euclid’s language, the notion
involved is congruence of the pieces. Their
credibility as representing Śulbasūtra thinking
derives from several factors taken together, each
of which may not be definitive. Firstly, the
statement of the geometric theorem in the
Śulbasūtra is strictly geometric, without reference
to numbers or computation. Number-free, cut-andfit geometry of areas is characteristic of many
constructions in the Śulbasūtra, some of which
are the same as the suggested proofs, especially
in the case of the square. Furthermore, long after
geometry had transcended visual reasoning, we
still find cut-and-fit proofs, only for the diagonal
theorem, as late as in the writings of Nīlakaha
and Jyehadeva. The robustness of the
transmission chain is a persuasive guarantee that
a commentator of say the 11th century who
ascribes his particular reading of a proposition to
Baudhāyana or Āpastamba is likely to be repeating
a line of argument that goes back a long way,
maybe even all the way to them.
69
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
For the square, the traditional proofs are
so elementary that Nīlakaha considered the
theorem to be self-evident. Fig. 3 shows two
variants. (The lengths of the side and the diagonal
are denoted by a and d respectively, merely for
the purpose of identification).
triangle) and gone on from there. It is not
impossible that the theorem reached him in its
general form, the very easy special case having
got mislaid somewhere, some time.
The final big advance in geometry was
Mādhava’s marriage of two themes from the past,
trigonometry and the idea of infinity, to create the
new discipline that, much later and in an alien
culture, came to be named calculus. We will catch
a glimpse of what is Indian in it in a later section
but, like Brahmagupta’s work, proper justice
cannot be done to it in a general review.
6 WHAT
Fig. 3. The diagonal theorem for the square
In the first variant for example, the square
on the diagonal, suitably drawn (the outer square),
is cut into eight congruent (isosceles) right
triangles, four of which fit together to form the
square on a side (the inner square), and that is all
of the proof. For the rectangle, the simplest proofs
are based on the same area-matching idea (there
is quite a collection of them) but a degree of
geometric imagination is also demanded. It seems
likely that the theorem was first formulated for
the square and then generalised; Baudhāyana’s
presentation is in that order. The strategy of first
dealing with the simplest special case and then
stretching it as far as it will go is very typical,
though it is not always as straightforward as it is
here; Brahmagupta’s theory of cyclic
quadrilaterals is a good illustration of how
powerful and technically subtle it could become.
It is easy to make these visual proofs
conform to Euclidean axiomatism; a good
definition of the square and the notion of congruent
right triangles will do the trick. One can wonder
if Euclid might have devised an equally trivial
proof if he had had the idea of first looking at the
square (which is not the same as the isosceles right
IS A PROOF?
Very few of the texts setting out original
results for the first time bother to give even an
indication of how they were obtained; the
Śulbasūtra are only the first illustration of this
tendency. Commentaries are a little better; they
explain difficult points, add their own
interpretations and so on but, with rare exceptions,
will disappoint those looking for sharply
enunciated and logically laid out demonstrations.
Even in the great masterpieces, terminology is
often not properly defined, conditions that apply
not explicitly stated (Āryabhaa, Brahmagupta, ...).
This gave rise to a widely shared belief that the
masters did not have communicable proofs for the
results they discovered and in the theories they
created or, worse, did not have universally agreed
criteria for what a proof should be. Modern
scholarship has begun to undo this
misunderstanding, mainly by recognising that the
real purpose of the canonical texts was to leave a
permanent record of knowledge newly gathered,
not to justify it; that was done in the parallel
pedagogic activity of face-to-face in-struction. In
the time-honoured Indian tradition of discussion
and debate (orality again), the ideal way of
learning was to have the master or someone of his
intellectaul progeny explain and clarify and
respond to questions, like Bhaspati with Indra
(see the next section).
70
INDIAN JOURNAL OF HISTORY OF SCIENCE
But even if we ignored this primary mode
of knowledge transmission, it would be naive to
imagine that mathematical knowledge – the best
of it far from being “self-evident to the intelligent”
– came out of informed guesswork or
computational astuteness alone (supplemented
perhaps by divine revelation?).7 It is a telling fact
that in India’s vast mathematical inheritance, there
are very very few results (and none that is genuinly
deep and original) which are plainly wrong, two
errors (on volumes of solids) by Āryabhaa (of all
people) being the most notorious. For mere
conjectures, this is an impossibly high rate of
success as we know in the context of modern
mathematics. Equally naive is to suppose that there
was no idea of what a proof should aim to be.
Mathematicians were not given to bringing logic
or philosophy into their writings – they barely
managed to get all their mathematics in – but it
would be absurd to think that they were ignorant
of or untouched by the spirited disputes and fine
hair-splittings of the various schools of philosophy/logic, a great deal of it about the
fundamentally epistemic question of how we know
something to be true. Indeed, the one
mathematician who did write about foundational
issues, Nīlakaha, shows himself to be very well
informed on such matters. It is time to lay to rest
the misconception that the best of gems from the
ocean of true and false knowledge were brought
up by any agency other than the exercise of
individual intelligence, svamati.
Āryabhaa has given us exactly one short
verse about where good mathematics comes from;
fortunately, his most perceptive interpreter,
Nīlakaha, has much more to say about how we
acquire knowledge and about how we are to know
it to be true. There is a background. Nīlakaha
was an exceptionally good mathematical
astronomer and his guru’s guru, Parameśvara, an
7
equally good observational astronomer who
revised the basic parameters governing planetary
motions, inherited ultimately from Āryabhaa, on
the ground that they no longer accounted for his
observations. In three books written at the end of
a long life championing his grand-guru’s work (in
an already full and fulfilling life), Nīlakaha
expresses himself freely on the enterprise of
acquiring and validating knowledge. The primary
instrumentality is that of our senses, even in
mathematics (it must be remembered that he was
partial to almost tactile, geometric, reasoning in
arithmetical and algebraic problems, by 2- and 3dimensional cut-and-fit methods). The ‘raw data’
are to be subjected to analysis and (tentative)
inferences drawn by means of our mental faculties,
to be then exposed to criticism by the
knowledgeable and taught to young students. If
the conclusions are at odds with the revelations
of sacred books (śruti) or the word of mortals
however influential (smti), the śruti-smti twins
are to be rejected. (It must also be remembered
that, as far as a rational world-view is concerned,
the thousand years after Āryabhaa was a period
of regressive Purāic ascendancy. A 9th century
astronomical text by one of his followers makes
fun of paurā ika-śruti on the subject of eclipses;
this battle is an old one).
Nīlakaha is critical of pure theory as,
according to him, theories are endless and
inconclusive. There is no room in his inductive
world view for any sort of axiomatism as indeed
there was not in Indian philosophical systems from
the time of the Buddha and the śrāmaa
movements; no first causes and first principles,
no unquestionable “postulates” and “common
notions”. And no strictly deductive proofs: if all
knowledge is contingent, how can it be otherwise
for metaknowledge, the knowledge that something
(anything) is or is not true? The principles of plane
The last but one stanza of the Āryabhaīya says: “From the ocean of true and false knowledge, by the grace of brahman, the best
of gems that is true knowledge has been brought up by me, by the boat of my own intelligence” (my emphasis; the Sanskrit word
is svamati). It must be added that, grammatically, this brahman is to be taken as the immanent spirit of the cosmos (neuter in
gender), not the male god of creation, Brahmā with a capital B.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
geometry and the rules of arithmetic are
trustworthy not because they derive from the
axioms of Euclid and of Peano (far into the future)
but because they work.
Nīlakaha practised what he preached.
Especially in the bhāya of the Gaita chapter of
the Āryabhaīya, his proofs of Āryabhaa’s
propositions are models of clarity and logical
completeness. As though in acknowledgement of
these newly advertised virtues, the word yukti for
proof (‘reasoned justification’ in the context)
began to displace the older term upapatti in
writings from Kerala. The significance of this shift
of emphasis in the organisation of proofs can be
gauged from the title, Yuktibhāā, of the
masterpiece written by his disciple Jyehadeva,
ostensibly as a mathematical guide to his
astronomy but much more than that in reality: a
textbook covering the high points of the
mathematics of the day, complete with detailed
proofs of all propositions.
The question of what constitutes a proof
outside a strictly axiomatic-deductive framework
has been a subject of intense debate over the last
century or so. Without speculating about
Nīlakaha’s credentials as an early intuitionist (it
has been suggested), it seems safe to say that the
Indian position in its maturity would have been
that it is a futile quest, endless and inconclusive.
As a matter of vyavahāra, practical considerations,
most mathematicians would have been happy to
leave the verdict on whether they had proved
something or not ultimately to the judgement of
their peers. Bhāskara II says at one place that a
piece of mathematics whose upapatti does not
meet with the approval of the learned assembly is
like rice without butter; the simile is interesting
in that he implies only that it is unpalatable.
Nīlakaha does not refer to the one
epistemic given that apparently had universal
acceptance among Indian mathematicians, which
is the rejection of the principle of the excluded
71
middle. The stance had major consequences, chief
among them the repudiation of the powerful proof
device of reductio ad absurdum, reduction to
absurdity, proof by contradiction. In the axiomatic
philosophy, to exclude the middle is to assert that
logic is bivalent: a proposition is either true or
not true (or, synonymously, false; it is good to be
pedantic because the issue is not free from
linguistic ambiguity). To prove that a proposition
is true, one starts by supposing that it is false and
shows that the axiom system (and the rules of
logic) within which the proposition is framed
necessarily leads to a contradiction which, by
bivalence, means that the proposition is true: a
proposition and its negation cannot both be false.
Among the earliest and simplest examples of its
use is Euclid’s (or perhaps someone else’s) proof
that the square root of 2 is not a rational number,
i.e., that it cannot be expressesd as a fraction. The
proof consists in showing that assuming that it is
a fraction (the negation of the proposition to be
proved) leads to a contradiction with a property
of fractions which says that they can always be
simplified by removing common factors from the
numerator and denominator. Now √2 is a ‘number’
the Śulbasūtra authors were familiar with
geometrically as the diagonal of the unit square
(which is also how the Greeks knew it) and
Baudhāyana gives a value for it as a fraction,
making it clear that it is not exact. The step to the
realisation that no fraction can be the exact value
of √2 was not taken until the very end when
Nīlakaha declares (very uncharacteristically,
with no yukti) that it cannot be determined.
The notion of irrationality appears
straightforward when formulated negatively (not
a fraction) but a satisfactory positive definition
(what is it then?) is subtle and was not even
attempted till the 19th century. Correspondingly,
all reasonably elementary proofs of irrationality
are by contradiction; it is in fact not easy to see
that even the sophisticated, more structurally
formulated, proofs are fully free from the taint of
72
INDIAN JOURNAL OF HISTORY OF SCIENCE
reductio at all levels of their total architecture. In
any case, one can easily understand that a culture
which refused to exclude the middle would have
had serious reservations about such proofs. Indian
scepticism about the philosophical utility of logical
bivalence goes far back, at least to the Buddha’s
reluctance to categorise what is and what is not,
well before Nāgārjuna formalised tetravalence:
that {yes, no, yes-and-no, neither-yes-nor-no}
exhaust all logical possibilities.
Bivalence requires (as indeed does
tetravalence) an unambiguous logical meaning to
be attached to linguistic negation: what does ‘not’
mean? Formally, to be able to usefully identify a
set S′ not having a particular property, one must
have a universal set U which circumscribes all the
possibilities one is willing to consider and of which
the set S which has the property is a sub-set; S′ is
the complement of S in U. In the irrationality
problem U is taken (tacitly in the early days, more
consciously nowadays) to be the set of all real
numbers, numbers which are the lengths of all
(geometric) straight lines (never mind how that is
defined). To see why the universe cannot be
dispensed with, we only have to note that all the
arithmetical steps of Euclid’s proof remain
unchanged if we ask whether √-2 can be expressed
as a fraction (once we extend numbers to include
negatives). It cannot be, but it is not a real number
either as we know. Indeed, one Indian
mathematician, very late, claimed that square roots
of negative numbers cannot exist since the square
of every number is positive. (The faint reflection
of bivalence that is seen here had the sanction of
some schools of logic but only in non-existence
proofs). The Buddha’s discourses did not, of
course, specify a universe of propositions,
encompassing as they did questions of life and
death, of cosmogony, metaphysics, ethics, etc.
Within such a wide and ill-defined horizon, it is
not obvious how to define negation in a formally
acceptable way.
8
The high point of Nīlakaha’s
engagement with irrationals is a remark-able
passage in his bhāya of the Āryabhaīya claiming
that the value of π (defined as the ratio of the
circumference of a circle to its diameter) can only
be given approximately because there is no single
unit of length that will measure both the
circumference and the diameter without a
remainder in at least one of them, in other words
that π cannot be a fraction. How did he know?
Once again, he does not say. My wishful
hypothesis is that he had some kind of a proof by
contradiction – technically it would have been
within his reach since the main tool in many
elementary proofs, the theory of continued
fractions, was part of the equipment of the Nila
school – which he kept to himself because it would
not have passed the test of philosophical approval.
A final comment on the subject of method.
Just as conspicuous as the absence of reductio is
the ubiquity of a style of reasoning that can be
called, in very general terms, recursive. A recursive
process is, loosely, one which results from the
iteration of an elementary process in which the
output of the nth step in the iteration is fed back
as the input of the (n + 1)th step.8 In this generality,
the idea is of wide applicability. Frits Staal in
particular has written extensively about the
presence of recursive patterns in the syntactic
structure of rituals, in the chants which accompany
them, and in grammatical constructions of various
sorts: padapāha and prātiśākhya in their role as
aids to memory (orality again) and, most
effectively, in the possibility of the repeated
application of nominal composition, potentially
without end. That alone makes it an essential
component of the Indianness of Indian
mathematics. Its mathematical avatars can,
naturally, be highly technical and we will have to
be content with a selective and qualitative
overview.
Logicians and computer theorists work with more precise definitions. The subject is of much current interest partly because
recursive algorithms are economical and effective in computation: maximal precision from minimal lines of code.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
The prototype recursive structure is the one
that defines that primordial object, the set of all
decimal (more generally, place-value or based)
numbers. The elementary process is division with
remainder. As we saw in section 3, the decimal
entry in the place of ones of a number N is obtained
as the remainder n1 when N is divided by 10: N =
10N1+n1; n2 in the place of tens as the remainder
when the quotient N1 is divided by 10: N1 = 10N2
+ n2; and so on. Having seen also that decimal
counting is as old as the earliest literary forms of
Vedic Sanskrit, we may well imagine that it might
have been the unacknowledged model for the
recursive elements in nonmathematical, in
particular linguistic, structures. And, as we shall
see in the next section, acknowledgement did
come eventually, from Bharthari.
Within the domain of mathematics itself,
the range of applications of the recursive idea is
amazingly diverse: definitions and constructions,
algorithms, expansions of functions as infinite
series, etc. – as is to be expected in India, the
dividing lines are not always very sharp – not to
mention its ultimate metamathematical
consummation as a method of proof, mathematical
induction. Depending on the problem, the
elementary process which will be iterated as well
as the initial input are matters of imagination and
choice. Apart from decimal counting itself, its
earliest use is as an algorithm for the square root
of a number which is not a square (a ‘number that
cannot be determined’) to arbitrary accuracy; there
is an explicit recursive formula for it in the
Bakhshali manuscript and it may well have been
used in deriving the Śulbasūtra approximation for
√2. Recursive logic runs through Āryabhaa’s
kuaka technique for the solution of the linear
indeterminate (Diophantine) equation as well as
Jayadeva’s solution (cakravāla) of Brahmagupta’s quadratic analogue of it. But it is in the
work of the Nila school that the method finds its
full power and glory under the very appropriate
name of saskāram: every iterative step is a
73
refining of the approximate output of the previous
step for better accuracy and, carried ad infinitum,
leads to the exact answer ‘in the limit’. The general
technique of saskāram was as indispensable in
the implementation of Mādhava’s programme as
the infinitesimal philosophy was to its conception.
The details are intricate and, regrettably, we have
to leave it at that. As an inadequate substitute, here
is a quick look at how it works for the square root.
The starting point in all saskāram
computations is an educated, consciously
approximate, first guess at the answer. For the
square root of a number N, the first guess is the
square root of the square number that is closest to
N; for example, if N is 109, the first guess for its
square root is 10. The exact answer is 10 + x for
some unknown x; all we know is that (10 + x)2 =
100 + 20x + x2 = 109 and that x is less than 1 since
112 is greater than 109. The quadratic equation for
x can be approximately solved without taking
square roots by neglecting x2 in comparison with
100 + 20x. Thus 10 is the first input and the
elementary process is the easy solution of a linear
equation which, in the first step, is 100 + 20x =
109, giving x = 9/20. The input in the next iteration
is x = 10 + 9/20 and so on; the process is repeated
as many times as needed for a given requirement
of precision (it is actually a very efficient
algorithm). An alternative viewpoint is to think
of the square root as a function of N and carry out
the iteration indefinitely, arriving thereby at an
infinite series expansion of √N for any N.
7. NAMING
AND KNOWING: THE PROBLEM OF
INFINITY
While language and grammar provided the
nourishing ambience in which mathematics
flourished, we must also note that the flow of ideas
was not all one way. The reductive power of the
decimal paradigm in making the unknowable
infinitude of numbers accessible through an
apprehension of a finite set of atomic numerals
had an appeal that went well beyond the purely
74
INDIAN JOURNAL OF HISTORY OF SCIENCE
mathematical, in particular to those who wrestled
with the unboundedness of language itself. The
concern about how to comprehend the infinite
potentiality of linguistic expressions through the
finite means at our command was already present
in very early writings on grammar; there is
Pata–jali’s fable about Bhaspati teaching
grammar to Indra by going over grammatical
expressions one by one and getting nowhere after
a thousand years of the gods, leading him to
conclude that they can be understood only by
means of some (finite number of) rules both
general and particular. General and particular rules
are precisely what go into the formation of
numbers. Several centuries later, in an act of
reciprocal generosity, decimal numbers –
inevitably, I would like to believe – became the
model for Bharthari (5th-6th century) for how
potentially unlimited structures in a language are
to be constructed from its elementary building
blocks. The perfectly named Vākyapadīya (“Of
Sentences and Words”) says: “Just as the grasping
of the first numbers is the means for the grasping
of other (or different) numbers, so it is with the
hearing of words”. (Note that words are heard).
The gloss, written by himself according to some
scholars, leaves no doubt about what he meant:
As the numbers beginning with one,
serving different purposes, are the means
of understanding numbers like hundred,
thousand, and so on, and are thought of
as constituent parts (avayava) of hundred
and so on, so the apprehension of a
sentence is based on the precise meaning
of words such as Devadatta, the
understanding of which is innate (or
inherent).9
Later on, he speaks of the atoms (au and
paramāu) of sound or speech (śabda) gathering
together, by their own capacity, like clouds: all
linguistic objects are formed by the agglomeration
of atoms of śabda, just as all numbers are formed
9
by the rule-bound coming together of atomic
numbers.
Bharthari was many things, linguist and
philosopher of language, epistemologist and
cognitive theorist, but a mathematician he was not.
Yet, the distinction he makes between the
cognition of the atomic numbers, ādya-sakhyā,
which is an innate capacity, and the compound
numbers formed by their clustering together is
accurate. The comprehension of these two classes
of numbers depends on quite distinct mental
processes. To know a compound number, we need
to apply the abstract arithmetical rules
underpinning the decimal system (the equivalent
of the ‘general rules’ of Pata–jali) no matter how
effortless it may appear in practice – that is what
the choice of a base such as 10 does. There cannot
be any rules for numbers less than the base and
that is why their cognition is an innate faculty –
and their names or symbols entirely arbitrary.
Concerns about how we ‘know’ a number
are very much older than Pata–jali, going as far
back as the genesis of the decimal system itself.
There is a line in a hymn to Agni from the early
Book IV of the gveda which equates the act of
counting to that of seeing and attributes the
capacity to do so to Agni, god of fire and light.
Louis Renou who first made the connection (later
taken up by Frits Staal) notes that khyā from which
sakhyāis derived is the verb root for seeing or
looking. Support for this identification comes from
Vedic words, still current, formed by attaching
prefixes other than sa to khyā all of which have
meanings derivable from illumination and/or
vision (examples: ākhyā, prakhyā, vikhyā, etc.).
And there are numerous passages in the gveda
that speak of Agni’s faculty of “comprehending
all things in this world minutely and correctly”,
arising out of his power to illumine. Numeracy
was a divine gift as language itself was; I like to
For mathematicians, the change in the value of the same numeral as a function of its position was of course not news. Nevertheless, the sense of novelty never seems to have worn off. Yuktibhāā has an enlightening passage in its first chapter titled
Sakhyāsvarūpam in which the value of a numeral in the place of ones is called its prakti and in higher places its vikti.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
imagine that the abundant presence of numbers in
the visionary poetry that the gveda otherwise is
is a celebration of this supreme achievement, that
of bringing to the world of mortals what had
belonged to the gods.10
It is not a matter of chance that the concept
of the infinite has been a background presence in
the paragraphs above. No linguist, certainly not
after Pāini, could have overlooked the infinite
generative power of grammar, the inexhaustibility
of grammatical expressions and the impossibility
of enu-merating them all, that Pata–jali’s fable
highlights, just as no one interested in numbers
and how they are built up could have ignored their
interminability. The prehistory of numerical
infinity as the unattainable limit of counting
numbers – not just as a proxy for some vague
metaphysical notion – is, correspondingly, much
older than that of the complementary idea of the
zero. The awareness, or at least the suspicion, that
numbers had no end was already very much in
the air in the earliest Vedic times. The highest
power of 10 in the gveda is 104 with the name
ayuta, but almost immediately afterwords, the
Yajurveda, in both its recensions, has long lists of
powers of 10 by name, pushing the highest power
up to 10 12 and further, in one list in the
Taittirīyasahitā, to 10 19. An even more
interesting set of nine lists has numbers increasing
in constant steps rather than exponentially,
generally beginning with small numerals and
ending around 100. There are two reasons that
make these lists interesting and it is rewarding to
look at them closely:
1. They start for instance like ekasmai svāhā,
tribhya svāhā, . . . (depending on the
particular list) and after reaching 100 (śatāya
svāhā), invariably end with a sort of coda:
sarvasmai svāhā (and so does one of the power
lists: after loka = 10 19, it is sarva). It is
10
75
impossible not to conclude that we are seeing
here the first explorations of the limits of
counting: no matter how one arranges the
count, there is no end to numbers; all one can
do is to stop at some point and let sarva be the
bridge that spans the totality of all numbers,
rather like the modern notational convenience
of · ··. The fascination with ever longer lists of
numbers kicked off by the Yajurveda
continued for a millennium and culminated in
the well-known episode from the Lalitavistāra
of Prince Siddhārtha’s pre-marital examination
in mathematics in which he astounds the world
with his knowledge and understanding of
impossibly (and unnamably) large numbers.
2. All numbers are cited by name. That meant
inventing new names for the higher powers of
10 (compoud number names being then taken
care of by grammar). They were taken from
the existing vocabulary, uas, samudra and so
on, without a direct numerical connotation, a
practice which quickly got out of hand and
led to the breakdown of a canonical association
of name and number. It would seem that the
pressure to find more and more names got to
the point where no one bothered to create a
system that could accommodate the logic of
the multiplicative process by which higher
powers were – in the mind – constructed. The
5th century Buddhist philosopher Vasubandhu
speaks in one of his works of 60 named powers
of 10 but actually gives only 52 of them,
explaining that the other 8 have been
vismtam, lost to memory. It may appear to be
no more than an amusing episode to us but
the problem of matching numbers and names
in a unique fashion was a serious one; it is in
fact possible to argue that the compulsive need
to have an unambiguous name for every
conceivable number was what came in the way
Parallels can be found in other places and at other times. Prometheus took from the gods not only fire but also the skill of
numbers. In India, at a later time when the Mahāyāna Buddhists were preoccupied with the endlessness of numbers and the
infinite multiplicity of the cosmos in space and time, the infinitely benevolent Avalokiteśvara, Bodhisatva of the infinitely
radiant Amitābha, was attributed Agni’s gift of acute vision.
76
INDIAN JOURNAL OF HISTORY OF SCIENCE
of an explicitly expressed recognition of their
inexhaustibility.
The reluctance to venture beyond the
nominal was bound to give way sooner or later.
Prince Siddhārtha in the Lalitavistāra adopts a
clever recursive technique to break free from the
tyranny of names: just describe the construction,
by a process of iterated powers of an already large
number, for example the number of grains of sand
in as many riverbeds as there are grains of sand in
one riverbed, and so on. The number of universes
of the Mahāyāna cosmogony occurs in the
discourse as does the word uncountable,
asakhyeya. Very soon afterwards (the Siddhārtha
fable is probably from the 3rd century AD), real
(not legendary) mathematicians had good
mathematical reasons to get to grips with infinity
in a more abstract setting. Both Āryabhaa and
Brahmagupta worked on a certain class of
equations called indeterminate and it turned out
that they had an unlimited number of different
solutions, an observation that was – finally! –
expressed through the unambiguous word ananta.
One would have thought that someone, a master
like Bhāskara II, would have come out and said
that the property of being ananta was a property
primarily of the counting numbers, but that did
not happen.
Ironically, the so-called classical period
also produced that most articulate champion of
the power of the spoken word, Bharthari, who
might well have been a contemporary of
Āryabhaa. At several places in the Vākyapadīya,
especially in those parts where he lays out the
general contours of his programme, the doctrine
is invoked according to which an abstract ‘thing’
is cognisable only by virtue of its being articulable.
Examples: “There is no cognition in this world
which does not involve the word (śabda). All
knowledge (jñāna) can be said to be intertwined
with the word”. Or: “All knowledge of what must
be done in this world (itikartavyata) is tied to the
word”. The synonymy that connects seeing and
knowing with naming has its roots, of course, in
the Vedic oral tradition; indeed, according to
Monier-Williams, khyā itself has the sense of ‘to
be named’ in the early Vedic corpus.
The Word never lost its power, even among
mathematicians: to name was to know, to cause
to exist. A millennium after Bharthari and two
millennia and a half after the Yajurveda, we find
the idea (ontological nominalism(?) if we must
have a name for it) reasserted by Jyehadeva in
the Yuktibhāā. Following the obligatory list of
the names of powers of 10 (up to 10 17),
Jyehadeva brings together name, existence and
the concept of infinity as a number all in one
marvellously revelatory line: “ Thus, if we endow
numbers with [repeated] multiplication [by 10] and
[the consequent] positional variation
(sthānabhedam), there is no end to the names of
numbers; hence we cannot know the numbers
themselves and their order” (my emphases, of
course).
The paradox is that the fear of the
unknowable did not stop Mādhava and those who
followed him from employing sequences of
numbers tending to infinity as a key input in the
creation of spectacularly new mathematics, that
of calculus on the circle. Jyehadeva explains how
it is done, with care and a sharp mathematical
sensibility. The relatively unexciting role of
infinity in the new mathematics is that the final
results themselves are in the form of infinite series
whose successive terms are described recursively
but quite explicitly; they were, in that sense,
‘known’. The far deeper one was that of producing
geometrical quantities, for instance an arc, as small
as we please, by dividing a finite arc by a number
as large as we please. It does not bother
Jyehadeva that the divisors could not be given
names and therefore could not be known; it was
enough that a sequence of unboundedly increasing
numbers did the job. Nevertheless, there must have
been a conflict, as evidenced by the trouble he
takes to explain how the nominal imperative was
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
subordinated to the demands of the mathematics
as it evolved. In the description of the actual
process of division, the divisor is first kept fixed
at parārddham (1017) which is the largest named
power of 10 in his list. It is then explained that the
method becomes increasingly (infinitely) more
accurate as 1017 is replaced by increasingly
(infinitely) higher powers of 10 and that
parārddham is just a notional number, used in the
description for definiteness of terminology. In
other words, Jyehadeva resolved the conflict by
first paying his dues to nominalism and then letting
the mathematics dictate what actually had to be
done. It is perhaps a measure of his unease that
the relevant passages are among the few where
the writing falls short of its usual clarity.
From a mathematical point of view, this
technique of ‘division by infinity’ is superior to
the early European attempts at defining
infinitesimals ab initio. Consequently, the Nila
calculus is sounder, conceptually and technically,
than that of Newton and Leibniz (though far more
narrowly circumscribed in scope). The distinction
may appear to be one of perception but it can be
traced, ultimately, to the two cultures’
understanding of the infinite. Central to calculus
is the notion of a limit which in turn needs the
notion of an infinite sequence – of numbers,
geometrical points, whatever – for its formulation.
For European mathematics, infinity as a
mathematically well-defined concept was not easy
to grasp, lacking as it did an intuitively natural
model for it such as the one provided by decimal
numbers. There are little asides in Newton’s
private notebooks expressing his prideful delight
at his discovery of “the doctrine recently
established for decimal numbers” – about three
millennia old by then in India – which he proposed
to use as a guide in the definition of infinite series.
8. NEW
DIRECTIONS
The final brilliant act in the story of Indian
mathematics – as the Nila school turned out to be
77
– marked the advent of many other breakout ideas
which moved it away from its traditional moorings
and towards what we can recognise now as modern
mainstream mathematics. It is not possible to do
justice to this new orientation in a survey like the
present one. From today’s perspective, its most
salient feature is that mathematics began to be
driven more and more by its own internal logic
and not only by possible applications,
astronomical or otherwise. The development of
calculus was not really called for by any
astronomical problem; methods for dealing with
them to the necessary numerical precision already
existed. And, within that general framework, the
technical apparatus deployed owed little to the
past, pointing instead to what was still to come in
Europe. To give one example, Mādhava derived
the sine and cosine series in a sequence of steps
which can be summarised in the modern
vocabulary of calculus as follows: i) set up the
differential equations satisfied by the functions,
ii) convert them into integral equations, and iii)
solve them by iteration (saskāram). Merely to
list these steps is to establish their credentials as
essential components of modern analysis as it
evolved in the 18th and 19th centuries.
Foundational advances outside the domain
of calculus narrowly defined include the
introduction of an abstract algebraic approach in
a problem of no great practical value (estimating
truncation errors in series) and the consequent
recognition of the need to answer (and, indeed,
ask) questions like: what is a polynomial?
Jyehadeva has an amazingly modern
characterisation of the natural numbers as being
defined primarily by their property of succession,
as close to axiomatic mathematics as India ever
came. This was in turn prompted by the need to
justify entirely new techniques of proof –
mathematical induction – where traditional styles
of demonstration failed. The idea of proof itself
underwent a subtle change primarily, as we saw
in section 6, through Nīlakaha’s reflections on
78
INDIAN JOURNAL OF HISTORY OF SCIENCE
the question of how we know mathematical
propositions (and astronomical doctrine, but that
does not concern us) to be true. Their impact is
evident in the new rigour that is a hallmark of
Yuktibhāā: proofs are unambiguous and
complete, with a noticeably formal, structured
character to them.
What is interesting for us in these signals
of a modernising mathematical mindset is that,
together, they constitute a reassertion of the
universal character of all good mathematics;
Indian mathematics became less Indian so to say.
The old order began to give way, gradually, under
the impact of Mādhava’s originality; it is not easy
for a modern reader who has ‘seen it all’ to grasp
at first sight what an extraordinary transformation
of the ethos of mathematics Yuktibhāā represents
when compared to earlier writings. First and
foremost is the notion of a function as defined over
its entire natural domain, the circle – calculus is
fundamentally about functions with some
continuity properties – not just as its values at a
few selected points as was the case earlier. Then
there are the other manifestations of the new
autonomy: the assured handling of infinite series
and infinity itself in the abstract, free of the
constraints of nominalism; the abandoning of the
search for geometric justifications of arithmetical
and algebraic identities; the first formulations of
abstract algebraic concepts and methods; the first
whiff of axiomatisation in the treatment of
numbers; the stress on formally presented proofs;
and so on, each of them a first intimation of a
radical move away from long-held habits of
thought. And all of them anticipated, in an
embryonic form, the evolutionary stages
mathematics passed through in Europe following
the Cartesian revolution;11 perhaps it is a truism
that every great advance in mathematics is a step
towards its universal core. It is another story that
11
creative mathematics was brought to an end in
Kerala – and in India – before these shoots could
take root and flourish, at about the time that Europe
was embarking on its great voyage of scientific
and mathematical discovery. The tragedy is that
the opening up in the 16th century of extensive
maritime channels of communication between
India and Europe, instead of inaugurating an era
of mutually stimulating exchanges, had the effect
of bringing the curtain down on mathematics in
India; what was a new beginning in one culture
turned out to be an end in the other.
9. CROSS-CULTURAL
CURRENTS?
The achievements of Mādhava and the
other Nila mathematicians and the new directions
they pointed to were subsumed in due course in
the explosive growth of mathematics in Europe
that began in the 17th century. And that leads to a
question the author have avoided so far, that of
the influences of different mathematical cultures
on one another or, in the spirit of my title: what is
non-Indian in Indian mathematics and, conversely,
what Indian traits can be identified in other
mathematical cultures? Much has been written,
and over a very long time now, about mathematical
give and take among different cultures, without
producing a proportionate degree of
enlightenment. The subject is a complex one
involving chronological priority, possible
mechanisms of contact and, most decisively,
assessments of commonalities in mathematical
themes and their working out. What makes a
rigorous and objective identification of
mathematical cross-currents across cultural
borders difficult, more so than in the case of, say,
linguistic borrowings, is the very universality and
immutability of the quarry: given propitious
circumstances, prepared and well-motivated
minds are as likely to unearth a mathematical “best
The one exception to this blanket statement about the Cartesian trigger is of course the axiomatic geometry of Euclid. But it was
only after Descartes that Euclid became the ideal to aspire to in other areas of mathematics (not to speak of other sciences and,
even, all human knowledge). An axiom system for numbers had to wait until Peano in the late 19th century.
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
of gems” at a given time and place as at any other;
we might even take that as an operational
definition of its universal identity and appeal.
In actuality, however, comparative history
of mathematics has not always been a shining
example of an objective and rigorous discipline.
Within the Indian context, it was perhaps natural
that colonial historians were not particularly
interested, especially from mid-19th century
onwards, in whatever data could be reliably
extracted or in what to do with them; the driving
urge was to derive everything from Classical
Greece.12 (It must also be said that some of the
earlier colonial historians – Colebrooke, Playfair,
Whish and some others – were much more openminded and did a commendable job with the sparse
material they had in hand). In the period following
the publication of the Śulbasūtra (by Thibaut,
1875), some Indianists dated Baudhāyana and
Āpastamba to the late centuries BC for no more
discernible reason than to preserve Euclid’s
primacy at least chronologically, regardless of the
philological evidence that both of them predated
Pāini. The Pythagorean theorem was a
particularly sensitive issue – and it must be
remembered that this was before the decipherment
of the Babylonian mathematical tablets and the
flutter they caused among historians of Classical
mathematics. The emphasis then shifted: is it
possible that the theorem of the diagonal came to
India from Babylonia?
The honest answer is that we still do not
know with certainty. Absolutely nothing is known
about any kind of contact between 18th century
BC Mesopotamia and the Vedic people who,
presumably, were making their first settlements
in northwestern India at that time. We do know
about extensive trade exchanges between
Mesopotamia and the Indus civilisation but that
was at its height 7 or 8 centuries earlier and had
12
79
come to an end. To confuse (or, maybe, clarify)
matters further, Indus artefacts have nothing
suggesting an acquaintance with the diagonal
theorem. In brief, and to come back where we
started, the known history of the Pythagorean
paradigm is very well accounted for if we are ready
to accept it as one of the great universal truths –
an element in the sense Proclus gave to Euclid’s
Elements – which more than one culture
discovered for itself. The same temporal mismatch
applies to the other great universal idea from
antiquity, place-value counting, and the
confusingly different ways it was implemented:
Indus Valley probably counted in base 8,
Babylonia, later, in base 60 and Vedic India, still
later (and orally), in base 10.
More generally, the readiness to accept
temporal antecedence (sometimes accompanied
by superficial similarities) as the determinant of
transmission is still a running motif in much
modern writing. Many examples can be given but
one that has found fairly wide acceptance will
suffice here. The background is the wholesale
importation into India of the Ptolemaic model of
planetary motion – as well as the supremely
important idea that such motions could be
described mathematically – during the so-called
Siddhānta period (the two or three centuries
preceding Āryabhaa), a true and well-documented
example of transmission. But Āryabhaa had no
use for the mathematics that (presumably) came
with it; he invented what he needed, mainly
trigonometry. Trigonometry was a bringing
together of two central elements of Śulbasūtra
geometry, the geometry of the circle and the
diagonal theorem, resulting in the key concept of
the half-chord as the natural linear object
associated to an arc. It is this object, the sine of an
angle, that helps turn much Greek geometry of
the circle into easy exercises in trigonometry, an
activity popular with some modern historians.
The urge was of course not confined to mathematics. To take an example not too far removed from it, it was seriously argued that
the notion of syllogism in Indian logic was dirctly descended from Aristotle, a “prepostorous notion” in the words of one
preeminent modern logician.
80
INDIAN JOURNAL OF HISTORY OF SCIENCE
Āryabhaa’s famous table of half-chords takes no
more than a day’s work, if that, once the ideas are
in place (though it must be said that some of the
auxiliary ideas are themselves very deep). Now,
Ptolemy in Alexandria had made an admirably fine
table of full chords, not half-chords, in the 2nd
century AD but, in the absence of the
simplifications that trigonometry afforded, it was
laborious work. It shares nothing mathematically,
in spirit or in technique, with Āryabhaa’s work,
except that they both dealt with arcs of circles (and
were useful to astronomers). Nevertheless, it is
an old habit, still vigorously alive in certain circles,
to say explicitly or through implication that
Āryabhaa’s table – and other facets of his
trigonometry – is to be traced to a direct influence
of Ptolemy, or to a predecessor of his whose
surviving writings are, regrettably or fortunately,
far too fragmentary for a meaningful comparison
to be feasible.
The Almagest of Ptolemy and the
Āryabhaīya are among the most thoroughly
analysed of the astronomical texts of antiquity. The
fact that that did not prevent conjecture from being
put forward as historical fact – at constant risk of
reasoned repudiation – is as good a measure as
any of how mere age or received opinion (the
Indian equivalents would be śruti and smti) can
trump the evidence of our eyes and ears and our
minds. (Nīlakaha should be alive now). In the
middle of the last century, Bartel van der Waerden,
a historian of Greek science and its relationship
with Mesopotamia and a very distinguished
mathematician, went so far as to declare in effect
that no great mathematical advance is made more
than once independently and that all subsequent
‘rediscoveries’ are derived causally from that
original epiphany. Insights revealed to the chosen,
once and never again, do not sound very much
like the unvarying outcome of a universally shared
faculty of the human mind. But it is a temptingly
attractive position for a historian to take as it
absolves him or her from the responsibility of
searching for evidence of anything beyond
chronological priority. If van der Waerden had not
died in the 1990s, when the availability of the
relevant Indian material in European languages
was still poor, he might have come to realise that
his dictum can cut both ways; few even among
the knowledgeable knew, even as recently as that,
what Mādhava had achieved two and a half
centuries before calculus was invented in Europe,
which is about the same time span as separated
Āryabhaa from Ptolemy (and at a time of
enormously easier communications). The surprise
is that the dictum, not always consciously
acknowledged or consistently applied, still has a
following among today’s historians, vastly better
informed, or at least with the means to be.
It is not impossible that new data may
come to light one day that will require a
reevaluation of the earliest phase of Indian
mathematical history and its linkages with the
outside world. The broad picture we have today,
especially from the time of Āryabhaa onwards,
is of a tradition that is comfortably self-sufficient,
perhaps overly conservative and even insular. How
else can one account for the fact that during the
first centuries AD, a time when Indo-Hellenic
cultural contacts were so intimate and over such a
broad front – from astronomy to architecture and
sculpture – Euclid, and Greek geometry generally,
was so assiduously ignored or rejected? Whatever
be the explanation, Indian science as a whole was
the poorer for it. It never produced an Archimedes
(who too seems to have been unheard of in India),
someone with a vision wide enough to include
natural science outside astronomy as a part of
mathematical thought; there was in fact no physics
in India at any time if we exclude, as we should,
purely metaphysical speculations (on atomism for
example).
The dynamics of the episodes of a reverse
flow, from India, is much better understood,
simply because the evidence in support is of much
better quality. Here again, possible very early
WHAT IS INDIAN ABOUT INDIAN MATHEMATICS?
cross-currents, for instance from the Indus
civilisation to Mesopotamia, remain uncharted. In
contrast, the much later spread of decimal
enumeration, first to Persia and the Abbasid
Caliphate and thence farther west, reaching Europe
in the 13th century, is very well documented. The
same wave carried Āryabhaan astronomy and the
mathematics that went with it, trigonometry in
particular, and played a huge part in the second
coming of Mesopotamian science in the following
centuries. Something similar happened with China
as well at about the same time, the agents of
transmission being Buddhist men of learning. That
the new arrival did not take to Chinese soil as well
as it did to the sands of Arabia, and declined in
step with the decline of Buddhism in that country
(in fact was abolished by royal decree at one
point), is perhaps another example of the role of
cultural predispositions in the welcome given to
alien knowledge. All of these instances illustrate
perfectly a conjunction of the conditions necessary
for a watertight case for transmission: priority,
established modes of communication, credible
documentation and, most critical of all, a
discernible impact on indigenous mathematics,
mutations of the mathematical DNA so to say.
It is legitimate to wonder now what the
great surge in European mathematics of the 17th
century – the birth of modern mathematics, quite
simply – might have owed to the rich but dying
Indian tradition. There is no doubt at all that the
‘new doctrine of numbers’, decimal enumeration
and its offshoots, had an absolutely decisive
impact, acknowledged by the mathematicians
themselves like Newton and Laplace if not always
by historians, most visibly in the maturing of
algebra as an autonomous discipline (as was also
the case in India). Brahmagupta’s work was known
in Islamic Spain but the direct inspiration for the
founder of modern number theory, Fermat, was
Diophantus of Alexandria (3rd-4th century AD),
who already had been translated into Latin by then.
Trigonometry, like decimal numbers, reached
Europe through Arabs, but later, and its adoption
81
into ‘pure’ mathematics (as opposed to, say,
cartography) was slower, perhaps because of the
hold of the geometric legacy of Euclid.
Mainstream geometry was almost exclusively
Euclidean, supplemented by the conics of
Apollonius. Above all, Descartes’ synthesis of
geometry and algebra through abstract graphical
reprsentations of functions, the most
transformative event of an eventful century, had
no Indian antecedents at all; no one on the banks
of the Nila drew the graph of the sine function.
It is against this background that we have
to judge whether word of Mādhava’s exploits
might have reached Europe and, if it did,
influenced the genesis of infinitesimal calculus
there. A positive answer would, of course, pass
the priority test. Channels of communication also
existed, especially with Kerala. But in the
mountains of paper the Portuguese generated, no
mathematical document with a link to Kerala has
so far been found (to my knowledge; it must be
added that two of their likely repositories were
destroyed: in Portugal as a result of the Lisbon
earthquake of 1755 and in Kerala in the burning
down of the great Jesuit library of Kochi by the
Dutch at almost exactly the time the young
Newton’s first ideas on calculus were
germinating). The more serious difficulty concerns
internal evidence; first indications – a proper study
remains a task for the future – are that the cultural
markers on the two avatars of calculus are as
distinctively individual as can be, given the
fundamental thematic unity. To mention only the
most prominent, there is no trace in Europe at all
of the Nila method of approaching the
infinitesimal limit by dividing by infinity. Its place
is taken by the subtle and original notion of the
derivative (Newton’s fluxion) or the differential
(Leibniz), without invoking directly the idea of
infinity at any point; presumably they were more
comfortable with quantities which are zero but not
quite zero than those which are infinite but not
quite infinite. Secondly, Europe was preoccupied
with many strictly local problems which needed
82
INDIAN JOURNAL OF HISTORY OF SCIENCE
only the differential half of calculus for their
solution, such as the construction of tangents to
curves and the determination of their (local) high
and low points (maxima and minima of functions);
the problems treated by Mādhava were global, of
which the differential half was just a preliminary
to the integral half (rectification, quadrature).
There are good historical reasons for these
variations in taste and choice. The generality of
the structural approach that Europe discovered in
Euclid’s geometry and carried forward through its
algebraisation contributed significantly to the
vision Newton and Leibniz had of calculus. They
recognised from the start that the concepts they
were working with were applicable to very general
functions and curves and said so explicitly; and it
is probably not an accident that Fermat, before
them, had made seminal contributions to the
founding of both analytic (Cartesian) geometry
and calculus. Add to this the role physical science,
in particular mechanics (which also was in the
process of being created as an exact science),
played in the work of both Fermat and Newton
and we have an idea of the ambition that the new
discipline was born with. The Nila mathematicians
came nowhere near that kind of breadth and scope,
not because of any inherent limitation in their
methods as we know now – division by infinity is
as versatile as the European infinitesimal limit –
but because they were content with solving the
handful of particular problems they were
interested in; it was a cultural thing. Alto-gether,
it is fair to say that if there is a convincing case
for any significant influence on European calculus
of what happened in distant Kerala, that case is
still to be made. The circle of ideas surrounding
the discipline of calculus is one of the great
universals of mathematics and it should surprise
no one that more than one gifted mind found its
way to it.
A final thought. Just as our mathematical
ancestors were creatures of their milieu, so are
those of us who study their achievements, as best
we can, of ours. In these days of scholarship by
assertion, that is all the more reason for us to heed
Nīlakaha’s advice about separating true
knowledge from false: trust our senses for the data
and our thinking minds for what we can deduce
from them; ignore all else, whether revealed in
the sacred texts (śruti) or handed down by mere
mortals (smti), when they go against facts and
logic.
BIBLIOGRAPHY
Sarasvati Amma, T.A. Geometry in Ancientand Medieval
India. Motilal Banarsidass, Delhi, 1979.
Bavare, Bhagyashree and Divakaran, P.P. Genesisand Early
Evolution of Decimal Enumeration: Evidence from
Number Names in gveda. IJHS 48.4 (2013):535.
Divakaran, P.P. The First Textbook of Calculus: Yuktibhāā.
J. Ind. Phil. 35 (2007): 417.
Divakaran, P.P. Notes on Yuktibhāā: Recursive Methods
in Indian Mathematics in C.S. Seshadri (Ed) Studies
in the History of Indian Mathematics, Hindustan Book
Agency, NewDelhi, 2010.
Datta, Bibhutibhushan and Singh, Avadhesh Narayan,
History of Hindu Mathematics Vol. I. Bharatiya Kala
Prakashan, Delhi, 1935, reprinted 2004.
Joseph, George Gheverghese, A Pasage to Infinity: Medieval
Indian Mathematics from Kerala and its Impact. Sage,
New Delhi, 2009.
Narasimha, Roddam. Epistemology and Language in Indian
Astronomy and Mathematics, J. Ind. Phil. (2007): 521.
Plofker, Kim. Mathematics in India. Princeton University
Press, Princeton, 2009.
Sen, S.N.and Bag, A.K. The Śulbasūtras of Baudhāyana,
Āpastamba, Kātyāyana and Mānava. Indian National
Science Academy, New Delhi, 1983.
Staal, Frits. Ritual and Mantras: Rules without Meaning.
Motilal Banarsidass, Delhi, 1990, Indian edition 1996.
Discovering the Vedas: Origins, Mantras, Rituals, Insights.
Penguin, New Delhi, 2008.
Tampuran, Ramavarma (Maru) and Akhilesvarayyar, A.R.
Yuktibhāā Part I-General Mathematics.
Mangalodayam, Trichur, 1948.
vander Waerden, B.L. Geometry and Algebrain Ancient
Civilizations. Springer, Berlin, 1983.